All-Electron Path Integral Monte Carlo (PIMC) Simulations of Warm Dense Matter: Application to Water and Carbon Plasmas

Transcription

1 All-Electron Path Integral Monte Carlo (PIMC) Simulations of Warm Dense Matter: Application to Water and Carbon Plasmas Kevin Driver and Burkhard Militzer Department of Earth and Planetary Science University of California, Berkeley APS March Meeting, Feb 28, 2012 Support provided by the NSF and UC Berkeley Computational support by NCAR, TAC, and TeraGrid

2 Motivation: need for new methods in the WDM regime ICF Hohlraum ICF ignition National Ignition Facility capsule ablation ICF Capsule Warm Dense Matter Planetary Cores Condensed Matter Warm dense matter physics occurs between condensed matter and plasma regimes. Effects of bonding, ionization, XC, and quantum degeneracy are all important. WDM methods are important for inertial confinement fusion (ICF) and planetary cores. Carbon is a promising ablator for ICF; We are working with LLNL on carbon EOS.

3 Motivation: Standard KS-DFT not sufficient for WDM ICF ignition NIF New Methods for WMD ICF capsule ablation PIMC, OF-DFT,? WDM Planetary Cores 100,000 K Standard KS-DFT (VASP, ABINIT) Standard KS-DFT intractable by T=10 ev due to large number of occupied orbitals. No reason to assume ground-state XC-functionals are valid for T > 0 ev. PIMC offers a highly accurate route to simulate WDM.

4 Overview of Path Integral Monte Carlo Method: PIMC is the quantum generalization of classical Lagrangian action principle. Thermal density matrix determines all thermodynamic properties of a many-body system. β H =[e ρ=e β H M M ] =[e M τ H ] (product property - time slicing) Imaginary time path integral at temperature T with time step tau. Sample paths from R to R' using action to accept/reject moves. R, R ' ; β)=... d R1... d R M 1 ρ( R, R' ; τ )... ρ( RM 1, R' ; τ) ρ( Z =Tr [ ρ], O =Z 1 Tr [ ρ O] Approximations: Many-body density approximated as pair density; valid for converged time step or T large. R, R ' ; β) ρ (r ij, r ' ij ;β) ρ( Fermion sign problem: positive and negative contributions to observable cancel. (R, R' ;β)>0 ρtrial Use trial nodes of free particle density matrix.

5 Do free particle nodes (FPN) work for first row elements? Thus far, PIMC with FPN has only been used to study hydrogen and helium plasmas. Is it possible to use the same method to study first row elements? 100% 2s High temperature ionization of a system: 1s ~0% 100% 100% Case 1: A fully ionized system (no bound states) => FPN vaild. Case 2: 1s is the only occupied state. No node required => FPN valid. Case 3: 1s and 2s fully occupied. Node between 1s-2s needed for correct shell structure. => FPN not valid. What if 2s and other states are just partially occupied? 2s <60% occupation 1s 100% occupation DFT occupation levels indicate 2s is ~60% occupied at T=250,000 K (carbon, water). Cross-validation between our PIMC and DFT results in the coming slides will show that FPN become valid near this condition for water and carbon. FPN accurate for T > 250,000 K for carbon and water plasmas.

6 Water: Pressure vs Temperature 1 Mbar structure P0 = ideal gas contributions 50 Mbar structure DFT-MD (VASP, PAW pseudopotentials, PBE XC functional). Gamma point calculations; at least 1500 ev plane wave energy cutoff. Use up to 1500 bands to converge partial occupancy to 10-4.

7 Water: Pressure vs Temperature 1 Mbar structure P0 = ideal gas contributions DFT-MD and classical Debye-Hückel plasma model. 50 Mbar structure

8 Water: Pressure vs Temperature 1 Mbar structure P0 = ideal gas contributions 50 Mbar structure DFT-MD and PIMC results overlap from 250, ,000 K. Cross-validation implies FPN sufficient and XC-functionals valid at high-t. PIMC results converge to the classical plasma (Debye-Hückel) model.

9 Carbon: Pressure vs Temperature 1 Mbar structure P0 = ideal gas contributions 50 Mbar structure DFT and PIMC results overlap from 250, ,000 K. Instantaneous AE-PP vs PAW-PP pressure comparison shows VASP-PAW PP fails at 2x106 K due to missing core excitation contributions.

10 Water and Carbon: Energy vs Temperature 1 Mbar structure E0 = ideal gas contributions 50 Mbar structure DFT and PIMC results overlap from 250, ,000 K.

11 Water and Carbon: Pair Correlation Functions PIMC and DFT-MD predict consistent structural properties. PCFs show sensitive temperature dependence.

12 Conclusions We have a PIMC method that produces accurate results for WDM. PIMC and DFT-MD together form a coherent equation of state from condensed matter to the plasma limit. PIMC pressures, internal energies, and pair correlation functions agree with DFT-MD in the range of 250, ,000 K. Results in press with PRL (out in March). Future Work: Validate more of the first row elements and combinations thereof.

13 Topics: Introduction to Monte Carlo Methods Pseudopotential generation Variational Quantum Monte Carlo Diffusion Quantum Monte Carlo Wavefunction optimization methods Path integral Monte Carlo QMC applications in geophysics Details: Scientists from geophysics, physics, materials science, chemistry and high-performance computing. Includes lectures and labs. $150 registration fee. Housing provided for non-local participants. Contact: Organizers: D. Ceperley, R. Cohen, E. de Sturler, J. Kim, B. Militzer, N. Sobh, and U. Ravaioli.

14 DFT-MD cell-size convergence for pair-correlation function

15 PIMC Energy vs. Time Step

16 Water: PIMC P vs. T for 24- and 6- atom cells